Is there anything like Meshmixer's “Attract Tool” in Blender?
This looks like a combination of Vertex Weights and a Shrinkwrap modifier.


On the object to be deformed, in its Data tab, Vertex Groups panel,
create a vertex group .. call it something like 'Shrinkwrap'
Assign vertices to it. Vertices can have a weight between 0 and 1 in the group, or not belong to the group at all.
One approach would be to assign only the vertices you want to deform
to the group, with weight 0....
Assign a Shrinkwrap modifier to the object, targeted on the object you want to shrink to,and affecting only the Vertex Group you've created.
Now in Weight Paint mode, you can hit 'Restrict' in the Tool tab > Options panel to protect vertices outside the Vertex Group from being affected..
.. and paint weight into the group. As you paint, the vertices will be drawn to the target.


There are numerous options .. in the modifier, and in the way you paint or assign weights to the vertices, (for instance, you can forget painting, and assign weights in Edit Mode) so really, it would be good to have a mooch in the docs and play with them, and if you get stuck and there's a particular way you'd like the tool-combination to behave, come back with a comment, and we can be more specific.